Difference between revisions of "Boundary"
From Encyclopedia of Mathematics
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The boundary of a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172501.png" /> of a given topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172502.png" /> is the set of points such that every neighbourhood of any point of it contains both points from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172503.png" /> and points from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172504.png" />. The accepted notations include <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172507.png" />. | The boundary of a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172501.png" /> of a given topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172502.png" /> is the set of points such that every neighbourhood of any point of it contains both points from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172503.png" /> and points from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172504.png" />. The accepted notations include <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017250/b0172507.png" />. | ||
| − | A synonym for the border of a [[ | + | A synonym for the border of a [[manifold]], such as the border of a simplex. |
| + | |||
| + | ====References==== | ||
| + | * J.L. Kelley, "General topology", Graduate Texts in Mathematics '''27''' Springer (1975) ISBN 0-387-90125-6 {{ZBL|0306.54002}} | ||
Revision as of 18:02, 10 October 2016
The boundary of a subspace 
of a given topological space 
is the set of points such that every neighbourhood of any point of it contains both points from 
and points from 
. The accepted notations include 
, 
, 
.
A synonym for the border of a manifold, such as the border of a simplex.
References
- J.L. Kelley, "General topology", Graduate Texts in Mathematics 27 Springer (1975) ISBN 0-387-90125-6 Zbl 0306.54002
How to Cite This Entry:
Boundary. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary&oldid=15128
Boundary. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary&oldid=15128
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article